# Degree of irreducible representation may be greater than exponent

## Contents

## Statement

We can have a finite group and an irreducible linear representation of the group over an algebraically closed field of characteristic zero, such that the degree of the irreducible representation is *greater* than the exponent of the group.

This is a non-constraint on the Degrees of irreducible representations (?) of a finite group.

## Related facts

### Similar facts

- Degree of irreducible representation need not divide exponent
- Square of degree of irreducible representation need not divide order
- Size of conjugacy class need not divide exponent

### Opposite facts

- Schur index of irreducible character in characteristic zero divides exponent, Schur index divides degree of irreducible representation: Thus, the Schur index of an irreducible character/representation divides both the degree of the representation
*and*the exponent. - Degree of irreducible representation divides group order
- Degree of irreducible representation divides order of inner automorphism group (i.e., the degree divides the index of the center)
- Degree of irreducible representation divides index of abelian normal subgroup
- Order of inner automorphism group bounds square of degree of irreducible representation

## Proof

### Example of big extraspecial groups

Consider an extraspecial group of order for any prime . The exponent of this group is either or . On the other hand, it admits a faithful irreducible representation of order .